Mixed Riemann-Hilbert boundary value problem with simply connected fibers

Abstract

We study the existence of solutions of mixed Riemann-Hilbert or Cherepanov boundary value problem with simply connected fibers on the unit disk Δ. Let L be a closed arc on ∂Δ with the end points ω−1,ω1 and let a be a smooth function on L with no zeros. Let {γξ}ξ∈∂Δ∖L˚ be a smooth family of smooth Jordan curves in C which all contain point 0 in their interiors and such that γω−1, γω1 are strongly starshaped with respect to 0. Then under condition that for each w∈γω±1 the angle between w and the normal to γω±1 at w is less than π10, there exists a Hölder continuous function f on Δ‾, holomorphic on Δ, such thatRe(a(ξ)‾f(ξ))=0 on Landf(ξ)∈γξ on ∂Δ∖L˚.